Supercloseness and Asymptotic Analysis of the Crouzeix–Raviart and Enriched Crouzeix–Raviart Elements for the Stokes Problem

For the Crouzeix–Raviart and enriched Crouzeix–Raviart elements of the Stokes problem, two pseudostress interpolations are designed and proved to admit a full one-order supercloseness with respect to the numerical velocity and the pressure, respectively. The design of these interpolations overcomes the difficulty caused by the lack of supercloseness of the canonical interpolations for the two nonconforming elements, and leads to an intrinsic and concise asymptotic analysis of numerical eigenvalues for the Stokes operator, which proves an optimal superconvergence of eigenvalues by the extrapolation algorithm. Meanwhile, an optimal superconvergence of postprocessed approximations for the Stokes equation is proved by use of this supercloseness. Finally, numerical experiments are tested to verify the theoretical results.